New generalized almost perfect nonlinear functions

APN (almost perfect non-linear) functions over finite fields of even characteristic are widely studied due to their applications to the design of symmetric ciphers resistant to differential attacks. This notion was recently generalized to GAPN (generalized APN) functions by Kuroda and Tsujie to odd characteristic p. They presented some constructions of GAPN functions, and other constructions were given by Zha et al. We present new constructions of GAPN functions both in the case of monomial and multinomial functions. Our monomial GAPN functions can be viewed as a further generalization of the Gold APN functions. We show that a certain technique used by Hou to construct permutations over finite fields also yields monomial GAPN functions. We also present several new constructions of GAPN functions which are sums of monomial GAPN functions, as well as new GAPN functions of degree p which can be written as the product of two powers of linearized polynomials. For this latter construction we describe some interesting differences between even and odd characteristic and also obtain a classification in certain cases.     

Power functions with low uniformity on odd characteristic finite fields

In this paper, we give some new low differential uniformity of some power functions defined on finite fields with odd characteristic. As corollaries of the uniformity, we obtain two families of almost perfect nonlinear functions in GF(3 ⁿ ) and GF(5 ⁿ ) separately. Our results can be used to prove the Dobbertin et al.’s conjecture.     

New families of perfect nonlinear polynomial functions

A new class of almost perfect nonlinear (APN) polynomial functions has been recently introduced. We give some generalizations of these functions and deduce new families of perfect nonlinear (PN) functions. We show that these PN functions are CCZ-inequivalent to the known perfect nonlinear functions.     

Fourier transforms on finite group actions and bent functions

Journal of Algebraic Combinatorics - Highly nonlinear functions (bent functions, perfect nonlinear functions, etc.) on finite fields and finite (abelian or nonabelian) groups have been studied in...     

On the classification of APN functions up to dimension five

We classify the almost perfect nonlinear (APN) functions in dimensions 4 and 5 up to affine and CCZ equivalence using backtrack programming and give a partial model for the complexity of such a search. In particular, we demonstrate that up to dimension 5 any APN function is CCZ equivalent to a power function, while it is well known that in dimensions 4 and 5 there exist APN functions which are not extended affine (EA) equivalent to any power function. We further calculate the total number of APN functions up to dimension 5 and present a new CCZ equivalence class of APN functions in dimension 6.     

Permutation polynomials with low differential uniformity over finite fields of odd characteristic

In this paper, we propose a construction of functions with low differential uniformity based on known perfect nonlinear functions over finite fields of odd characteristic. For an odd prime power q, it is proved that the proposed functions over the finite field Fq are permutations if and only if q≡3(mod 4).     

几乎完全非线性多项式函数间的EA等价性

简述了几乎完全非线性(APN)多项式函数的研究现状,讨论了两类几乎完全非线性多项式函数间的扩张仿射(EA)等价性,给出了验证EA等价的一般方法.     

完美非线性映射的一类构造

在分组密码中 ,为了抗差分攻击 ,需要完美非线性映射 .利用有限域 Zp上的广义 Bent函数和不可约多项式 ,给出了完美非线性映射的一类构造 .     

Equivalences of quadratic APN functions

The following conjecture due to Y. Edel is affirmatively solved: two quadratic APN (almost perfect nonlinear) functions are CCZ-equivalent if and only if they are extended affine equivalent.     

Galois环和Z/(m)环上完全非线性函数的性质

本文把完全非线性函数推广到了有限Abel群上,利用特征谱讨论了Z/(m)上Bent函数与GF(pe)上bent函数以及完全非线性函数定义之间的关系;给出Galois环与Z/(m)上最佳线性逼近的特征谱表示,得到完全非线性函数在某种程度上能抵抗最佳线性逼近攻击的结论;并给出一种Galois环与Z/(m)环上完全非线性函数的构结方法.     

On differential uniformity and nonlinearity of functions

We present results related to vectorial plateaued functions and mappings whose derivatives are 2^s-to-1 functions. The results in this note generalize facts about almost perfect nonlinear and almost bent functions. We investigate the connection between plateaued and 2^s-to-1 functions. We show that functions which are both plateaued and differentially uniform give rise to partial difference sets.     

Binary linear codes from vectorial boolean functions and their weight distribution

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding et al. recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contain the all-one codeword, and determine their weight distribution. Second, we investigate a subcode of any linear code mentioned above and consider its parameters. When the vectorial Boolean function is a perfect nonlinear function or a Gold function in odd dimension, we can completely determine the weight distribution of this subcode. Besides, our linear codes have larger dimensions than the ones by Ding et al.’s generic construction.     

Relative difference sets,graphs and inequivalence of functions between groups

For cryptographic purposes, we want to find functions with both low differential uniformity and dissimilarity to all linear functions and to know when such functions are essentially different. For vectorial Boolean functions, extended affine equivalence and the coarser Carlet–Charpin–Zinoviev (CCZ) equivalence are both used to distinguish between nonlinear functions. It remains hard to tell when CCZ equivalent functions are EA‐inequivalent. This paper presents a framework for solving this problem in full generality, for functions between arbitrary finite groups. This common framework is based on relative difference sets (RDSs). The CCZ and EA equivalence classes of perfect nonlinear (PN) functions are each derived, by quite different processes, from equivalence classes of splitting semiregular RDSs. By generalizing these processes, we obtain a much strengthened formula for all the graph equivalences which define the EA equivalence class of a given function, amongst those which define its CCZ equivalence class. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 260–273, 2010     

Authentication Schemes from Highly Nonlinear Functions

We construct two families of authentication schemes using highly nonlinear functions on finite fields of characteristic 2. This leads to improvements on an earlier construction by Ding and Niederreiter if one chooses, for instance, an almost bent function as the highly nonlinear function.     

一些完全非线性函数的原像分布值

确定完全非线性函数的原像分布值,是决定和分析完全非线性函数以及构造相应线性码的公开问题和重要课题之一.本文讨论了完全非线性函数的原像分布所满足的基本方程的求解问题,完全解决了该方程当m=5以及m=6时的情形.     

G-Perfect nonlinear functions

Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with G-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of G-perfect nonlinear functions, both -valued and -valued. Our final constructions demonstrate G-perfect nonlinear planar permutations (from to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions.      

Almost perfect commutative rings

Almost perfect commutative rings R are introduced (as an analogue of Bazzoni and Salce's almost perfect domains) for rings with divisors of zero: they are defined as orders in commutative perfect rings such that the factor rings $R/Rr$ are perfect rings (in the sense of Bass) for all non-zero-divisors$r\in R$. It is shown that an almost perfect ring is an extension of a T-nilpotent ideal by a subdirect product of a finite number of almost perfect domains. Noetherian almost perfect rings are exactly the one-dimensional Cohen–Macaulay rings. Several characterizations of almost perfect domains carry over practically without change to almost perfect rings. Examples of almost perfect rings with zero-divisors are abundant.     

Multi-sequences with Almost Perfect Linear Complexity Profile and Function Fields over Finite Fields

Sequences with almost perfect linear complexity profile defined by Niederreiter (1997, Lecture Notes in Computer Science, Vol. 304, pp. 37–51, Springer-Verlag, Berlin/New York) are quite important for stream ciphers. In this paper, we investigate multi-sequences with almost perfect linear complexity profile and obtain a construction of such multi-sequences by using function fields over finite fields. Some interesting examples from this construction are presented to illustrate our construction.     

Characterizing Translations on Groups by Cosets of Their Subgroups

On a group, constant functions and left translations by group elements map left cosets into left cosets for every subgroup. We determine classes of groups for which this property of preserving cosets characterizes constants and translations, e.g., finite non-abelian groups that are perfect, partitioned, primitive, or generated by elements of prime order p. For certain classes of groups we construct other coset-preserving functions, in particular, power endomorphisms and functions defined in terms of the subgroup lattice.     

A family of skew Hadamard difference sets

In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley-Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new perfect nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these perfect nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley-Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new perfect nonlinear functions has applications in cryptography, coding theory, and combinatorics.